In Call of Duty: Ghosts, the new Extinction mode is kind of like if you were to mix Zombies from Black Ops 2 and Survival from Spec Ops in Modern Warfare 3. In my opinion it’s one of the cooler new aspects in Call of Duty: Ghosts.
There are multiple ways to unlock Extinction Mode in Call of Duty: Ghosts.
How to Unlock Extinction Mode
Method 1
Playing in the Campaign on Regular difficulty or higher, complete the first four missions. These should be Ghost Stories, Brave New World, No Man’s Land, and Struck Down.
Method 2
Reach level 5 in any of the online Multiplayer modes